Chapter 10

# 10.8 Evans Root Locus Construction Rule #6: Angles of Departures/Arrivals at Complex Poles/Zeros

This Rule deals with angles of departure (arrival) from/to complex poles (zeros).

Rule 6: If $G(s)$ has a pole p of multiplicity r, then r branches of the root locus depart from p. The angle of departure of these root loci from p are described by this equation:

 $\theta_{dep}$= $\frac{ \angle [G(s)(s-p)^r]_{s=p}+(2k+1) \cdot 180^{\circ}}{r}$  $k=0,1,2,...,r-1$                                                                                                                                                                                 Equation 10-13

Similarly, if $G(s)$ has a zero z of multiplicity r, then r branches of the root locus arrive at z. The angle of arrival of these root loci to z are described by this equation:

 $\theta_{arr} = \frac{-\angle[\frac{G(s)}{(s-z)^r}]_{s=z}+(2k+1) \cdot 180^{\circ}}{r} k=0,1,2,...,r-1$ Equation 10-14

See examples in the next section for illustration of this rule.